Theorem cocanfo 38477

Description: Cancellation of a surjective function from the right side of a composition. (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Assertion

Ref	Expression
cocanfo	⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐺 = 𝐻)

Proof of Theorem cocanfo

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 781	. . . . . 6 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹))
2	1	fveq1d 6884	. . . . 5 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑦) = ((𝐻 ∘ 𝐹)‘𝑦))
3		simpl1 1210	. . . . . . 7 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐹:𝐴–onto→𝐵)
4		fof 6793	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
5	3, 4	syl 18	. . . . . 6 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐹:𝐴⟶𝐵)
6		fvco3 6982	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑦) = (𝐺‘(𝐹‘𝑦)))
7	5, 6	sylan 592	. . . . 5 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑦) = (𝐺‘(𝐹‘𝑦)))
8		fvco3 6982	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑦) = (𝐻‘(𝐹‘𝑦)))
9	5, 8	sylan 592	. . . . 5 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑦) = (𝐻‘(𝐹‘𝑦)))
10	2, 7, 9	3eqtr3d 2805	. . . 4 ⊢ ((((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) ∧ 𝑦 ∈ 𝐴) → (𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)))
11	10	ralrimiva 3156	. . 3 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → ∀𝑦 ∈ 𝐴 (𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)))
12		fveq2 6882	. . . . . 6 ⊢ ((𝐹‘𝑦) = 𝑥 → (𝐺‘(𝐹‘𝑦)) = (𝐺‘𝑥))
13		fveq2 6882	. . . . . 6 ⊢ ((𝐹‘𝑦) = 𝑥 → (𝐻‘(𝐹‘𝑦)) = (𝐻‘𝑥))
14	12, 13	eqeq12d 2778	. . . . 5 ⊢ ((𝐹‘𝑦) = 𝑥 → ((𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)) ↔ (𝐺‘𝑥) = (𝐻‘𝑥)))
15	14	cbvfo 7294	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑦 ∈ 𝐴 (𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
16	3, 15	syl 18	. . 3 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → (∀𝑦 ∈ 𝐴 (𝐺‘(𝐹‘𝑦)) = (𝐻‘(𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
17	11, 16	mpbid 235	. 2 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥))
18		eqfnfv 7026	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) → (𝐺 = 𝐻 ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
19	18	3adant1 1148	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) → (𝐺 = 𝐻 ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
20	19	adantr 486	. 2 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → (𝐺 = 𝐻 ↔ ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = (𝐻‘𝑥)))
21	17, 20	mpbird 260	1 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐺 = 𝐻)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078   ∘ ccom 5663   Fn wfn 6532  ⟶wf 6533  –onto→wfo 6535  ‘cfv 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545

This theorem is used by: (None)